Quick solution:
Use a CS2000 or CS2100 Smart PLL (requires a crystal oscillator), use a cheap micro to talk to it via I2C at power-up, set it to multiply 48kHz by 250x, output 12 MHz, done. You'll have to read the datasheet for details, but it doesn't look too complicated. Here's the product page. This is the chip XMOS uses in their eval boards to generate audio clocks. The resulting clock won't be as clean as a VCXO wrt jitter, but that chip is pretty good.
Note the two clock inputs: one is a local "low jitter timing reference" (ie, a crystal oscillator) and the other "50Hz to 30MHz" is your 48kHz signal. This is a bit counter-intuitive but you should not use a 12MHz oscillator in order to avoid a low frequency beat between the reference and the output, see datasheet page 13. A 10MHz oscillator should work fine.
CS2000 acts both as a stand-alone clock generator and as a configurable PLL ; CS2100 is PLL only ; CS2200 is clock generator only. There are more chips without I2C that won't suit your needs, and there are various versions of each, it's a bit confusing.
CS2000 will probably be the best fit if you want to run the soundcard both with GPS synchro (using the PLL) and without GPS (using the stand-alone clock generator mode).
It is certainly much simpler than a VCXO or the huge TI chip you linked, it's a cheap SO package, it doesn't have noise-inducing sub-nanosecond output rise times, uses one single 3.3V supply, draws only 60mW so you can power it from the soundcard... Quite practical.
The board will probably be small enough to fit inside your USB soundcard box, which means you'll just add a 48kHz input to the soundcard, maybe drill a hole in the back. If you don't want ground loops, you can use TOSLINK optical TX/RX to transmit the 48kHz signal.
Putting it inside the box avoids injecting 12MHz clock into the soundcard via a coax sticking out of it and soldered on to the PCB. Maybe you'll find a nice ground area in exactly the right spot to terminate the shield properly, maybe not and you'll end up with an ugly pigtail. It can cause EMI issues. If the coax ground is connected to a noisy place the clock will be jittery, also the coax ground could make a ground loop with your clock generator and introduce hum or noise, etc.
Over-engineered solution:
After you mod it, you and all the other musicians are going to be married to that Behringer soundcard and unable to change it. So I'll challenge the question and go with another approach, which will not require modding the soundcard, and will work with any soundcard. Also it will be far away from the ADC so whatever noise it introduces will matter a lot less. I'll try to make a list of options...
The easiest way to slave a soundcard to a clock is if it has a wordclock or SPDIF input but this one does not.
Note I'll use ASRC as the acronym for asynchronous sample rate converter/conversion. This is the process of resampling an audio stream into a new sample clock domain that is not synchronized to the original clock.
Software ASRC
You're probably not the only guy doing this, so there must be a way to sync the PC using NTP and use that to resample the audio streams. If the software can't do ASRC using NTP or similar network protocol, then perhaps it can do it using another audio input as sample clock reference.
In this case, I'd use a PLL to generate 128Fs (6.144MHz) from your 48kHz GPS clock (see below) then use WM8804 or any other SPDIF transmitter chip to generate SPDIF synchronized to that and carrying all-zero audio data. Or, if you have any kind of audio equipment that has a wordclock input and a SPDIF output, you can also use that to generate SPDIF synced to your GPS, and you don't even have to design any hardware.
Once you have a SPDIF stream synchronized to your GPS, input that into the PC using the motherboard's SPDIF input, and find a software plugin which can do ASRC between two audio streams and synchronize your recording audio stream to that.
You can record both the original audio from the soundcard and the resampled one too, that can be useful if the resamping plugin doesn't sound good.
Hardware ASRC
If software refuses to cooperate, next solution would be hardware.
The idea is to route the audio through a hardware asynchronous sample rate converter (ASRC) like SRC4392 from TI or equivalent from another manufacturer, and feed that chip with a clock synchronized to your GPS.
Something like: route the audio to the PC's SPDIF output, then through a SRC4392 which has SPDIF I/O, then back to a SPDIF input on the PC. This may require a cheap adaptive USB-to SPDIF-output interface, or maybe this.
However the SRC4392 still requires a clock, so you still need ths CS2300.
Summary: Using a hardware or software ASRC as I proposed above would allow you to use a dirty or jittery clock, so your PLL implementation won't really matter as long as it outputs the correct frequency. Also it is outside the soundcard, so it'll work with any soundcard.
